Inductive Definitions in Bounded Arithmetic: A New Way to Approach P - - PowerPoint PPT Presentation

CTFM, Feb 19, 2013, Tokyo Institute of Technology

Inductive Definitions in Bounded Arithmetic: A New Way to Approach P vs. PSPACE

Naohi Eguchi

Mathematical Institute, Tohoku University, Japan

Introduction 1/2

• Purpose in computational complexity:

Find limits of realistic computations.

• Theoretically: Comparing different notions about

computational complexity, e.g. P ̸=? NP

Introduction 1/2

• Purpose in computational complexity:

Find limits of realistic computations.

• Theoretically: Comparing different notions about

computational complexity, e.g. P ̸=? NP

• Difficult: to compare complexity classes directly.

= ⇒ Machine-independent logical approaches.

• This talk: new Bounded Arithmetic

characterisations of P and PSPACE. (P ⊆ NP ⊆ PSPACE, P ̸=? PSPACE)

Introduction 2/2 In finite model theory (N. Immermann et al.)

• 1. P is captured by monotone inductive definitions.
• 2. PSPACE is captured by non-monotone inductive

definitions.

Introduction 2/2 In finite model theory (N. Immermann et al.)

• 1. P is captured by monotone inductive definitions.
• 2. PSPACE is captured by non-monotone inductive

definitions. Can 1 or 2 be formalised in bounded arithmetic?

• to understand what is the most essential principle

in P- or PSPACE-computations.

• to find new aspects of the relationship between P

and PSPACE.

Overview 1/4

Inductive definition (monotone case) Example of inductive definition: N is the smallest set containing 0 closed under x → x + 1.

Overview 1/4

Inductive definition (monotone case) Example of inductive definition: N is the smallest set containing 0 closed under x → x + 1. More precisely: Define an operator F : V → V by x ∈ F (X) :⇔ x = 0 ∨ ∃y ∈ X(x = y + 1).

Overview 1/4

Inductive definition (monotone case) Example of inductive definition: N is the smallest set containing 0 closed under x → x + 1. More precisely: Define an operator F : V → V by x ∈ F (X) :⇔ x = 0 ∨ ∃y ∈ X(x = y + 1). See:

• N is the least fixed point of F :

F (N) ⊆ N, ∀X ⊆ V [F (X) ⊆ X → N ⊆ X]

• The least fixed point exists since F is monotone:

X ⊆ Y ⇒ F (X) ⊆ F (Y ).

Overview 2/4

Inductive definition (general case) F : V → V ; x ∈ F (X) :⇔ x = 0 ∨ ∃y ∈ X(x = y + 1).    F 0 := ∅ F α+1 := F (F α) F γ := ∪

α<γ F α

(γ : limit)

Overview 2/4

Inductive definition (general case) F : V → V ; x ∈ F (X) :⇔ x = 0 ∨ ∃y ∈ X(x = y + 1).    F 0 := ∅ F α+1 := F (F α) F γ := ∪

α<γ F α

(γ : limit) See:

• ∃α0 < #P(V ) such that

F α0+1 = F (F α0) = F α0.

• N = Fα0.

Overview 3/4

Inductive definition (finite case) F : S → S (#S < ω)

• There does not always exist m < ω such that

F m+1 = F (F m) = F m.

• However ∃k ≤ 2#S, ∃l > 0 such that

∀n ≥ l, F k+n = F n.

Overview 3/4

Inductive definition (finite case) F : S → S (#S < ω)

• There does not always exist m < ω such that

F m+1 = F (F m) = F m.

• However ∃k ≤ 2#S, ∃l > 0 such that

∀n ≥ l, F k+n = F n.

Overview 3/4

Inductive definition (finite case) F : S → S (#S < ω)

• There does not always exist m < ω such that

F m+1 = F (F m) = F m.

• However ∃k ≤ 2#S, ∃l > 0 such that

∀n ≥ l, F k+n = F n. Note:

• Choice of k and l is not unique.
• But F n plays a role similar to the least fixed

point like in infinite case.

Overview 4/4

Connection to time-complexity Suppose:

• 1. A function f(x) is computable in T (x) steps.
• 2. TAPEl denotes the tape description at the lth

step in computing f(x); TAPE0 = B i1 · · · i|x| B · · · B (x = i1 · · · i|x| (input), i1, . . . , i|x| ∈ {0, 1})

Overview 4/4

Connection to time-complexity Suppose:

• 1. A function f(x) is computable in T (x) steps.
• 2. TAPEl denotes the tape description at the lth

step in computing f(x); TAPE0 = B i1 · · · i|x| B · · · B (x = i1 · · · i|x| (input), i1, . . . , i|x| ∈ {0, 1})

Overview 4/4

Connection to time-complexity Suppose:

• 1. A function f(x) is computable in T (x) steps.
• 2. TAPEl denotes the tape description at the lth

step in computing f(x); TAPE0 = B i1 · · · i|x| B · · · B (x = i1 · · · i|x| (input), i1, . . . , i|x| ∈ {0, 1}) Then

• TAPET (x)+1 = TAPET (x).
• This gives rise to (finite) inductive definition!

Formalising computations 1/2 f is computable ⇔ ∃ program to compute f

• Σ0

1-formula

This gives rise to: Def Let Φ: a set of formulas ⊆ Σ0

1 & f: a function.

f is Φ-definable in T if ∃A(⃗ x, y) ∈ Φ such that

• 1. All free variables in A(⃗

x, y) are indicated.

• 2. n = f( ⃗

m) ⇔ N | = A( ⃗ m, n) for ∀ ⃗ m, n ∈ N.

• 3. T ⊢ ∀⃗

x∃!yA(⃗ x, y).

Formalising computations 1/2 f is computable ⇔ ∃ program to compute f

• Σ0

1-formula

This gives rise to: Def Let Φ: a set of formulas ⊆ Σ0

1 & f: a function.

f is Φ-definable in T if ∃A(⃗ x, y) ∈ Φ such that

• 1. All free variables in A(⃗

x, y) are indicated.

• 2. n = f( ⃗

m) ⇔ N | = A( ⃗ m, n) for ∀ ⃗ m, n ∈ N.

• 3. T ⊢ ∀⃗

x∃!yA(⃗ x, y).

Formalising computations 2/2 Classical facts:

• 1. f: primitive recursive ⇔ f: Σ0

1-definable in IΣ1.

(Parsons ’70, Mints ’73, Buss ’86 and Takeuti ’87)

Formalising computations 2/2 Classical facts:

• 1. f: primitive recursive ⇔ f: Σ0

1-definable in IΣ1.

(Parsons ’70, Mints ’73, Buss ’86 and Takeuti ’87)

• 2. f ∈ FP ⇔ f: Σb

1-definable in S1 2.

(Buss ’86)

• The start of bounded-arithmetic

characterisations of complexity classes. Note: By G¨

• del’s incompleteness theorem, not all

the computable functions are definable in any reasonable system.

Inductive definitions in 2nd order arithmetic

• Inductive definition can be axiomatised in 2nd
• rder arithmetic in the most natural way.

Fact

• 1. Π1

0-MID0 = Π1 1-CA0.

(MID: Monotone Inductive definition)

• 2. Π1

0-MID0 = Π0 1-MID0 ⊊ Π0 2-ID0 ⊊

Π0

3-ID0 ⊊ · · · .

Inductive definitions in 2nd order arithmetic

• Inductive definition can be axiomatised in 2nd
• rder arithmetic in the most natural way.

Fact

• 1. Π1

0-MID0 = Π1 1-CA0.

(MID: Monotone Inductive definition)

• 2. Π1

0-MID0 = Π0 1-MID0 ⊊ Π0 2-ID0 ⊊

Π0

3-ID0 ⊊ · · · .

• Finitary inductive definition can be axiomatised in

2nd order bounded arithmetic.

Foundations of 2nd order bounded arithmetic 1/3 Languages of 2nd order bounded arithmetic:

• 1. 0, S, + and ·.
• 2. ⌊ x

2 ⌋, |x| = ⌈log2(x + 1)⌉ and |X|.

Importantly x#y = 2|x|·|y| is not included. Intuition:

• 1. X, Y, Z · · · ∈ <N{0, 1}.
• 2. |X| = l if X ≡ i0i1 · · · il−1 & ij ∈ {0, 1}.
• 3. j ∈ X ⇔ ij = 1 if X ≡ i0i1 · · · il−1.

Foundations of 2nd order bounded arithmetic 1/3 Languages of 2nd order bounded arithmetic:

• 1. 0, S, + and ·.
• 2. ⌊ x

2 ⌋, |x| = ⌈log2(x + 1)⌉ and |X|.

Importantly x#y = 2|x|·|y| is not included. Intuition:

• 1. X, Y, Z · · · ∈ <N{0, 1}.
• 2. |X| = l if X ≡ i0i1 · · · il−1 & ij ∈ {0, 1}.
• 3. j ∈ X ⇔ ij = 1 if X ≡ i0i1 · · · il−1.

Foundations of 2nd order bounded arithmetic 2/3 Def (ΣB

1 -formulas)

• 1. ΣB

0 = ΠB 0 : the set of formulas containing only

bounded number quantifiers ∃x ≤ t.

• 2. ∃ ⃗

X(| ⃗ X| ≤ ⃗ t ∧ φ( ⃗ X)) ∈ ΣB

n+1 if φ ∈ ΠB n.

Foundations of 2nd order bounded arithmetic 2/3 Def (ΣB

1 -formulas)

• 1. ΣB

0 = ΠB 0 : the set of formulas containing only

bounded number quantifiers ∃x ≤ t.

• 2. ∃ ⃗

X(| ⃗ X| ≤ ⃗ t ∧ φ( ⃗ X)) ∈ ΣB

n+1 if φ ∈ ΠB n.

Def (Bit-comprehension axiom) ∀x∃X≤x s.t. ∀j < x(j ∈ X ↔ φ(j)) (∃X≤x · · · denotes ∃X(|X| ≤ x ∧ · · · )) Note: ∪

n∈N ΣB n ⊆ ∆0 1(exp) ⊆ Σ0 1 by definition.

Foundations of 2nd order bounded arithmetic 3/3

2nd order arith. 2nd order BA 1st order ob- jects elements of N ≤ p(|x|) 2nd order ob- jects f : N → N f : p(|x|) → {0, 1} typical classes

• f formulas

Σ1

n

ΣB

n

(p: polynomial)

Foundations of 2nd order bounded arithmetic 3/3

2nd order arith. 2nd order BA 1st order ob- jects elements of N ≤ p(|x|) 2nd order ob- jects f : N → N f : p(|x|) → {0, 1} typical classes

• f formulas

Σ1

n

ΣB

n

(p: polynomial)

Def Vn := BASIC + ΣB

n-COMP.

ΣB

n-COMP: BCA with φ restricted to ΣB n.

Thm (Zambella ’96) f ∈ FPΣP

n ⇔ f: ΣB

n+1-definable in Vn+1.

Formalising inductive definitions Def ∀x, ∃X≤x, ∃Y ≤x s.t. Y ̸= ∅ and

• 1. ∀j < x(P ∅

ϕ(j) ↔ j = 0) (i.e. P ∅ ϕ = ∅)

• 2. ∀Z∀j < |Z|(P S(Z)

ϕ

(j) ↔ φ(j, P Z

ϕ ) ∧ j < x)

• 3. ∀j < x(P X+Y

ϕ

(j) ↔ P Y

ϕ (j))

(P X

ϕ : fresh predicate, S: binary successor X → X + 1)

Recall:

• 1. F 0 = ∅
• 2. F m+1 = F (F m)
• 3. ∃k ≤ 2#S, ∃l ̸= 0 s.t. F k+l = F l

Formalising inductive definitions Def ∀x, ∃X≤x, ∃Y ≤x s.t. Y ̸= ∅ and

• 1. ∀j < x(P ∅

ϕ(j) ↔ j = 0) (i.e. P ∅ ϕ = ∅)

• 2. ∀Z∀j < |Z|(P S(Z)

ϕ

(j) ↔ φ(j, P Z

ϕ ) ∧ j < x)

• 3. ∀j < x(P X+Y

ϕ

(j) ↔ P Y

ϕ (j))

(P X

ϕ : fresh predicate, S: binary successor X → X + 1)

Recall:

• 1. F 0 = ∅
• 2. F m+1 = F (F m)
• 3. ∃k ≤ 2#S, ∃l ̸= 0 s.t. F k+l = F l

Formalising inductive definitions Def ∀x, ∃X≤x, ∃Y ≤x s.t. Y ̸= ∅ and

• 1. ∀j < x(P ∅

ϕ(j) ↔ j = 0) (i.e. P ∅ ϕ = ∅)

• 2. ∀Z∀j < |Z|(P S(Z)

ϕ

(j) ↔ φ(j, P Z

ϕ ) ∧ j < x)

• 3. ∀j < x(P X+Y

ϕ

(j) ↔ P Y

ϕ (j))

(P X

ϕ : fresh predicate, S: binary successor X → X + 1)

Recall:

• 1. F 0 = ∅
• 2. F m+1 = F (F m)
• 3. ∃k ≤ 2#S, ∃l ̸= 0 s.t. F k+l = F l

Capturing P and PSPACE Def ΣB

0 -IDEF:

Axiom of inductive definition for φ ∈ ΣB

0 .

Thm 1 Every f ∈ FP is ΣB

1 -definable in V0 + ΣB 0 -IDEF.

Thm 2 Every f ∈ FPSPACE is ΣB

1 -definable in

V0 + ΣB

0 -IDEF.

Overview 4/4

Connection to time-complexity Suppose:

• 1. A function f(x) is computable in T (x) steps.
• 2. TAPEl denotes the tape description at the lth

step in computing f(x); TAPE0 = B i1 · · · i|x| B · · · B (x = i1 · · · i|x| (input), i1, . . . , i|x| ∈ {0, 1}) Then

• TAPET (x)+1 = TAPET (x).
• This gives rise to (finite) inductive definition!

Proof of Theorem 2 Suppose: f ∈ FPSPACE. ∃p: poly { f(x) is computable in 2p(|x|)steps |TAPEX| ≤ p(|x|) See: TAPEX → TAPEX+1: ΣB

0 .

By (ΣB

0 -IDEF) ∃K, ∃L s.t. TAPEK+L = TAPEL.

See: TAPEL must be in the accepting state. So f(x) = y ⇔ ∃X≤p(|x|), ∃Y ≤p(|x|) TAPEX+Y = TAPEY ∧ y = output(TAPEY ) Hence f is ΣB

1 -definable in V0 + ΣB 0 -IDEF.

Proof of Theorem 2 Suppose: f ∈ FPSPACE. ∃p: poly { f(x) is computable in 2p(|x|)steps |TAPEX| ≤ p(|x|) See: TAPEX → TAPEX+1: ΣB

0 .

By (ΣB

0 -IDEF) ∃K, ∃L s.t. TAPEK+L = TAPEL.

See: TAPEL must be in the accepting state. So f(x) = y ⇔ ∃X≤p(|x|), ∃Y ≤p(|x|) TAPEX+Y = TAPEY ∧ y = output(TAPEY ) Hence f is ΣB

1 -definable in V0 + ΣB 0 -IDEF.

Proof of Theorem 2 Suppose: f ∈ FPSPACE. ∃p: poly { f(x) is computable in 2p(|x|)steps |TAPEX| ≤ p(|x|) See: TAPEX → TAPEX+1: ΣB

0 .

By (ΣB

0 -IDEF) ∃K, ∃L s.t. TAPEK+L = TAPEL.

See: TAPEL must be in the accepting state. So f(x) = y ⇔ ∃X≤p(|x|), ∃Y ≤p(|x|) TAPEX+Y = TAPEY ∧ y = output(TAPEY ) Hence f is ΣB

1 -definable in V0 + ΣB 0 -IDEF.

Proof of Theorem 2 Suppose: f ∈ FPSPACE. ∃p: poly { f(x) is computable in 2p(|x|)steps |TAPEX| ≤ p(|x|) See: TAPEX → TAPEX+1: ΣB

0 .

By (ΣB

0 -IDEF) ∃K, ∃L s.t. TAPEK+L = TAPEL.

See: TAPEL must be in the accepting state. So f(x) = y ⇔ ∃X≤p(|x|), ∃Y ≤p(|x|) TAPEX+Y = TAPEY ∧ y = output(TAPEY ) Hence f is ΣB

1 -definable in V0 + ΣB 0 -IDEF.

Inflationary inductive definition Can Theorem 1 be sharpen?: Thm 1 Every f ∈ FP is ΣB

1 -definable in

V0 + ΣB

0 -IDEF.

Inflationary inductive definition Can Theorem 1 be sharpen?: Thm 1 Every f ∈ FP is ΣB

1 -definable in

V0 + ΣB

0 -IDEF.

Def An operator F is inflationary if X ⊆ F (X). Note: Inflationary inductive definition can be reduced monotone one over FOL. (Gurevich-Shelah ’86)

Inflationary inductive definition Can Theorem 1 be sharpen?: Thm 1 Every f ∈ FP is ΣB

1 -definable in

V0 + ΣB

0 -IDEF.

Def An operator F is inflationary if X ⊆ F (X). Note: Inflationary inductive definition can be reduced monotone one over FOL. (Gurevich-Shelah ’86) We can define: Def ΣB

0 -IIDEF: a restriction of ΣB 0 -IDEF to

inflationary inductive definition.

Results Thm 1 (sharpened) f ∈ FP if and only if ΣB

1 -definable in V0 + ΣB 0 -IIDEF.

(⇐ =) Reduce ΣB

0 -IIDEF to V0 + ΣB 1 -IND = V1.

Recall: Thm (Zambella ’96) f ∈ FP ⇔ f: ΣB

1 -definable in V1.

Conjecture Conjecture ΣB

0 -IDEF can be reduced to W1 1.

(W1

1: 3rd order extension of V1)

Thm (Skelley ’06) f ∈ FPSPACE ⇔ f is ΣB

1 -definable in W1 1.

(ΣB

1 : 3rd order extension of ΣB 1 )

Corollary of Conjecture f ∈ FPSPACE ⇔ f is ΣB

1 -definable in

V0 + ΣB

0 -IDEF.

Conjecture Conjecture ΣB

0 -IDEF can be reduced to W1 1.

(W1

1: 3rd order extension of V1)

Thm (Skelley ’06) f ∈ FPSPACE ⇔ f is ΣB

1 -definable in W1 1.

(ΣB

1 : 3rd order extension of ΣB 1 )

Corollary of Conjecture f ∈ FPSPACE ⇔ f is ΣB

1 -definable in

V0 + ΣB

0 -IDEF.

Conclusion

• Finite model-theoretic characterisations of P and

PSPACE can be reformulated by inductive definitions in bounded arithmetic.

• P vs. PSPACE can be reduced to inflationary vs.

non inflationary inductive definitions.

• PSPACE can be discussed about without using

3rd order notions. – V1 (2nd order) corresponds to P. – W1

1 (3rd order) corresponds to PSPACE.

Thank you for your attention! Speaker is generously supported by the John Templeton Foundation.